Nowadays a lot of printed matter is produced carrying a reproduction of a black and white or colour image. A large part of these prints are produced using offset printing but in office and home environment a lot of prints are made using relatively small printing apparatuses.
Possible types of printers are typically laser printers using an electrographic process, thermal printers and inkjet printers.
Older printers were only capable of recording one type or size of dot, a dot of colorant was either absent or present. These types use so-called binary printing processes.
Recently apparatuses are capable of reproducing several sizes or densities of dots for each colorant. Such a printer uses a multilevel process. An example of this type of printer is an inkjet printer capable of jetting drops of different sizes or a variable number of drops on to of each other onto a substrate resulting in different dot sizes. Another method is making use of different inks having the same colour but different densities (e.g. light and dark magenta inks or black and grey inks).
Also a combination of the two methods (different densities/different drop sizes) is used (U.S. Pat. No. 5,975,671 by Spaulding et al.).
Printing processes seldom behave linearly, i.e. there is no linear relationship between the electronic level of the pixels to be applied and the optical density of the printed pixel. In order to obtain a good representation of the image to be printed the printing process has to be calibrated in advance.
By calibration of a printing process we mean the calculation and application of a gradation compensation curve for each of the colorants, to bring the gradation to a standard and stable state.
Following considerations regarding a multilevel inkjet printing process can be made. Reference is made to FIG. 1.
In a K-level printing process, K basic tone levels exist. These basic tone levels may arise from printing with dots of multiple sizes, from using inks with different densities but substantially the same hue, or from a combination of both. We indicate the K different levels by L1, L2, . . . , LK. The resulting basic tone levels are indicated by T1, T2, . . . , TK, i.e. a patch of tone Ti is formed by laying down level Li at each pixel in the patch.
Intermediate tone levels are created by a multilevel halftone procedure.
From the point of view of graininess, it is preferable to form a tone level situated between Ti and Ti+1, by a mixture of pixels having level Li and pixels having level Li+1 only.
The printing process is naturally divided into several regimes:                the regime where pixels of level L1=white are mixed with pixels of level L2,        the regime where pixels of level L2 are mixed with pixels of level L3,        etc.        
By a regime we understand a part of the tone scale printed with a mixture of a specific set of (two) levels.
To take a specific example, consider an inkjet printing process able to deliver two drop sizes. In the first half of the tone scale small dots are placed with white spaces in between until all pixels are filled with the small dots. In the second half of the tone scale, the small dots are replaced at some pixels by large dots. At the darkest tone, all pixels are filled with large dots. FIG. 1 shows the density as a function of the tone level for such a process. At the border of the two regimes (i.e. at the tone T2) we see an un-smooth behaviour of the gradation, a nod as illustrated in FIG. 1.
The density behaviour between T1 and T2 is substantially linear if we increase the percentage of pixels filled with small dots in a linear way with the tone level. The density behaviour between T2 and T3 is also substantially linear although it may deviate from linearity at the darker tones due to dot overlaps (depicted by the dotted line in FIG. 1).
The nod at T2 is noticeable as an abrupt change or a contour in a slowly varying image portion. Although the print process is continuous at the point, its gradation is not smooth and our eyes is are sensitive to it.
In the calibration process, we want to bring the process to a standard state, characterised by a predefined smooth gradation curve. Since the process is un-smooth itself, the only way to bring it to a smooth gradation curve is to apply an un-smooth correction. The current method aims to model the gradation of the printing process by a piecewise smooth curve and to correct the process with a piecewise smooth gradation-correction curve to bring it to a predefined smooth target curve.
Traditional calibration methods try to model the measured data with an overall smooth curve, to produce an overall smooth gradation-correction curve. This will never yield satisfactory results if the printing process is un-smooth itself.